DOCSLIB.ORG

CS 6347 Lecture 4 Markov Random Fields

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
CS 6347 Lecture 4 Markov Random Fields CS 6347 Lecture 4 Markov Random Fields (a.k.a Undirected Graphical Models) Bayesian Network • A Bayesian network is a directed graphical model that captures independence relationships of a given probability distribution – Encodes local Markov independence assumptions that each node is independent of its non-descendants given its parents – Corresponds to a factorization of the joint distribution 푝 푥1, … , 푥푛 = 푝(푥푖|푥푝푎푟푒푛푡푠(푖)) 푖 2 Limits of Bayesian Networks • Not all sets of independence relations can be captured by a Bayesian network – 퐴 ⊥ 퐶 | 퐵, 퐷 – 퐵 ⊥ 퐷 | 퐴, 퐶 • Possible DAGs that represent these independence relationships? 퐴 퐷 퐵 퐷 퐵 퐶 퐶 퐴 3 Markov Random Fields (MRFs) • A Markov random field is an undirected graphical model – Undirected graph 퐺 = (푉, 퐸) – One node for each random variable – One potential function or "factor" associated with cliques, 퐶, of the graph – Nonnegative potential functions represent interactions and need not correspond to conditional probabilities (may not even sum to one) 4 Markov Random Fields (MRFs) • A Markov random field is an undirected graphical model – Corresponds to a factorization of the joint distribution 1 푝 푥 , … , 푥 = 휓 (푥 ) 1 푛 푍 푐 푐 푐∈퐶 ′ 푍 = 휓푐(푥푐) ′ ′ 푥1,…,푥푛 푐∈퐶 5 Markov Random Fields (MRFs) • A Markov random field is an undirected graphical model – Corresponds to a factorization of the joint distribution 1 푝 푥 , … , 푥 = 휓 (푥 ) 1 푛 푍 푐 푐 푐∈퐶 ′ 푍 = 휓푐(푥푐) ′ ′ 푥1,…,푥푛 푐∈퐶 Normalizing constant, 푍, often called the partition function 6 An Example 퐴 퐵 퐶 1 • 푝 푥 , 푥 , 푥 = 휓 (푥 , 푥 )휓 (푥 , 푥 ) 휓 (푥 , 푥 ) 퐴 퐵 퐶 푍 퐴퐵 퐴 퐵 퐵퐶 퐵 퐶 퐴퐶 퐴 퐶 • Each potential function can be specified as a table as before 푥퐴 = 0 푥퐴 = 1 1 1 휓퐴퐵 푥퐴, 푥퐵 = 1 0 7 The Ising Model • Mathematical model of ferromagnets • Each atom has an associated spin that is biased by both its neighbors in the material and an external magnetic field – Spins can be either +1 or -1 +1 −1 +1 −1 – Edge potentials capture the local interactions +1 +1 +1 +1 – Singleton potentials capture the external field +1 −1 +1 −1 1 푝 푥 = exp ( ℎ 푥 + 퐽 푥 푥 ) −1 −1 −1 +1 푉 푍 푖 푖 푖푗 푖 푗 푖∈V 푖,푗 ∈퐸 8 Independence Assertions • Instead of d-separation, we need only consider separation: – If 푋 ⊆ 푉 is graph separated from 푌 ⊆ 푉 by 푍 ⊆ 푉, (i.e., all paths from 푋 to 푌 go through 푍) then 푋 ⊥ 푌 | 푍 – What independence assertions follow from this MRF? 퐴 퐷 퐵 퐶 9 Independence Assertions 퐴 퐵 퐶 1 푝 푥 , 푥 , 푥 = 휓 푥 , 푥 휓 (푥 , 푥 ) 퐴 퐵 퐶 푍 퐴퐵 퐴 퐵 퐵퐶 퐵 퐶 • How does separation imply independence? • Show that 퐴 ⊥ 퐶 | 퐵 10 Independence Assertions • In particular, each variable is independent of all of its non-neighbors given its neighbors – All paths leaving a single variable must pass through some neighbor • If the joint probability distribution, 푝, factorizes with respect to the graph 퐺, then 퐺 is an I-map for 푝 • If 퐺 is an I-map of a positive distribution 푝, then 푝 factorizes with respect to the graph 퐺 – Hamersley-Clifford Theorem 11 BNs vs. MRFs Property Bayesian Networks Markov Random Fields Conditional Factorization Potential Functions Distributions Product of Conditional Normalized Product of Distribution Distributions Potentials Cycles Not Allowed Allowed Partition Potentially NP-hard to 1 Function Compute Independence d-Separation Graph Separation Test 12 Moralization • Every Bayesian network can be converted into an MRF with some possible loss of independence information – Remove the direction of all arrows in the network – If 퐴 and 퐵 are parents of 퐶 in the Bayesian network, we add an edge between 퐴 and 퐵 in the MRF • This procedure is called "moralization" because it "marries" the parents of every node 퐴 퐷 ? 퐵 퐶 13 Moralization • Every Bayesian network can be converted into an MRF with some possible loss of independence information – Remove the direction of all arrows in the network – If 퐴 and 퐵 are parents of 퐶 in the Bayesian network, we add an edge between 퐴 and 퐵 in the MRF • This procedure is called "moralization" because it "marries" the parents of every node 퐴 퐷 퐴 퐷 퐵 퐶 퐵 퐶 14 Moralization 퐶 퐶 퐴 퐵 퐴 퐵 • What independence information is lost? 15 Factorizations • Many factorizations over the same graph may represent the same joint distribution – Some are better than others (e.g., they more compactly represent the distribution) – Simply looking at the graph is not enough to understand which specific factorization is being assumed 퐴 퐷 퐵 퐶 16 Factor Graphs • Factor graphs are used to explicitly represent a given factorization over a given graph – Not a different model, but rather different way to visualize an MRF – Undirected bipartite graph with two types of nodes: variable nodes (circles) and factor nodes (squares) – Factor nodes are connected to the variable nodes on which they depend 17 Factor Graphs 1 푝 푥 , 푥 , 푥 = 휓 (푥 , 푥 )휓 (푥 , 푥 ) 휓 푥 , 푥 퐴 퐵 퐶 푍 퐴퐵 퐴 퐵 퐵퐶 퐵 퐶 퐴퐶 퐴 퐶 퐴 휓퐴퐵 휓퐴퐶 퐵 퐶 휓퐵퐶 18 Conditional Random Fields (CRFs) • Undirected graphical models that represent conditional probability distributions 푝 푌 푋) – Potentials can depend on both 푋 and 푌, typically only the observed variables are considered in the model 1 푝 푌 푋) = 휓 (푥 , 푦 ) 푍(푥) 푐 푐 푐 푐∈C ′ 푍 푥 = 휓푐(푥푐, 푦푐) 푦′ 푐∈C 19 Log-Linear Models • CRFs often assume that the potentials are log-linear functions 휓푐 푥푐, 푦푐 = exp(푤 ⋅ 푓푐(푥푐, 푦푐)) where 푓푐 is referred to as a feature vector and 푤 is some vector of feature weights • The feature weights are typically learned from data • CRFs don’t require us to model the full joint distribution (which may not be possible anyhow) 20 Conditional Random Fields (CRFs) • Binary image segmentation – Label the pixels of an image as belonging to the foreground or background – +/- correspond to foreground/background – Interaction between neighboring pixels in the image depends on how similar the pixels are • Similar pixels should preference having the same spin (i.e., being in the same part of the image) 21 Conditional Random Fields (CRFs) • Binary image segmentation – This can be modeled as a CRF where the image information (e.g., pixel colors) is observed, but the segmentation is unobserved – Because the model is conditional, we don’t need to describe the joint probability distribution of (natural) images and their foreground/background segmentations – CRFs will be particularly important when we want to learn graphical models 22 Low Density Parity Check Codes • Want to send a message across a noisy channel in which bits can be flipped with some probability – use error correcting codes 휓퐴 휓퐵 휓퐶 푦1 푦2 푦3 푦4 푦5 푦6 휙1 휙2 휙3 휙4 휙5 휙6 푥1 푥2 푥3 푥4 푥5 푥6 • 휓퐴, 휓퐵, 휓퐶 are all parity check constraints: they equal one if their input contains an even number of ones and zero otherwise • 휙푖 푥푖, 푦푖 = 푝 푦푖 푥푖 , the probability that the th bit was flipped during transmission23 Low Density Parity Check Codes 휓퐴 휓퐵 휓퐶 푦1 푦2 푦3 푦4 푦5 푦6 휙1 휙2 휙3 휙4 휙5 휙6 푥 푥 푥3 푥 푥 푥 1 2 4 5 6 • The parity check constraints enforce that the 푦’s can only be one of a few possible codewords: 000000, 001011, 010101, 011110, 100110, 101101, 110011, 111000 • Decoding the message that was sent is equivalent to computing the most likely codeword under the joint probability distribution 24 Low Density Parity Check Codes 휓퐴 휓퐵 휓퐶 푦1 푦2 푦3 푦4 푦5 푦6 휙1 휙2 휙3 휙4 휙5 휙6 푥 푥 푥3 푥 푥 푥 1 2 4 5 6 • Most likely codeword is given by MAP inference arg max 푝 푦|푥 푦 • Do we need to compute the partition function for MAP inference? 25 .
Load more

Recommended publications
  • Methods of Monte Carlo Simulation II
    Methods of Monte Carlo Simulation II Ulm University Institute of Stochastics Lecture Notes Dr. Tim Brereton Summer Term 2014 Ulm, 2014 2 Contents 1 SomeSimpleStochasticProcesses 7 1.1 StochasticProcesses . 7 1.2 RandomWalks .......................... 7 1.2.1 BernoulliProcesses . 7 1.2.2 RandomWalks ...................... 10 1.2.3 ProbabilitiesofRandomWalks . 13 1.2.4 Distribution of Xn .................... 13 1.2.5 FirstPassageTime . 14 2 Estimators 17 2.1 Bias, Variance, the Central Limit Theorem and Mean Square Error................................ 19 2.2 Non-AsymptoticErrorBounds. 22 2.3 Big O and Little o Notation ................... 23 3 Markov Chains 25 3.1 SimulatingMarkovChains . 28 3.1.1 Drawing from a Discrete Uniform Distribution . 28 3.1.2 Drawing From A Discrete Distribution on a Small State Space ........................... 28 3.1.3 SimulatingaMarkovChain . 28 3.2 Communication .......................... 29 3.3 TheStrongMarkovProperty . 30 3.4 RecurrenceandTransience . 31 3.4.1 RecurrenceofRandomWalks . 33 3.5 InvariantDistributions . 34 3.6 LimitingDistribution. 36 3.7 Reversibility............................ 37 4 The Poisson Process 39 4.1 Point Processes on [0, )..................... 39 ∞ 3 4 CONTENTS 4.2 PoissonProcess .......................... 41 4.2.1 Order Statistics and the Distribution of Arrival Times 44 4.2.2 DistributionofArrivalTimes . 45 4.3 SimulatingPoissonProcesses. 46 4.3.1 Using the Infinitesimal Definition to Simulate Approx- imately .......................... 46 4.3.2 SimulatingtheArrivalTimes . 47 4.3.3 SimulatingtheInter-ArrivalTimes . 48 4.4 InhomogenousPoissonProcesses. 48 4.5 Simulating an Inhomogenous Poisson Process . 49 4.5.1 Acceptance-Rejection. 49 4.5.2 Infinitesimal Approach (Approximate) . 50 4.6 CompoundPoissonProcesses . 51 5 ContinuousTimeMarkovChains 53 5.1 TransitionFunction. 53 5.2 InfinitesimalGenerator . 54 5.3 ContinuousTimeMarkovChains .
    [Show full text]
  • 12 : Conditional Random Fields 1 Hidden Markov Model
    10-708: Probabilistic Graphical Models 10-708, Spring 2014 12 : Conditional Random Fields Lecturer: Eric P. Xing Scribes: Qin Gao, Siheng Chen 1 Hidden Markov Model 1.1 General parametric form In hidden Markov model (HMM), we have three sets of parameters, j i transition probability matrix A : p(yt = 1jyt−1 = 1) = ai;j; initialprobabilities : p(y1) ∼ Multinomial(π1; π2; :::; πM ); i emission probabilities : p(xtjyt) ∼ Multinomial(bi;1; bi;2; :::; bi;K ): 1.2 Inference k k The inference can be done with forward algorithm which computes αt ≡ µt−1!t(k) = P (x1; :::; xt−1; xt; yt = 1) recursively by k k X i αt = p(xtjyt = 1) αt−1ai;k; (1) i k k and the backward algorithm which computes βt ≡ µt t+1(k) = P (xt+1; :::; xT jyt = 1) recursively by k X i i βt = ak;ip(xt+1jyt+1 = 1)βt+1: (2) i Another key quantity is the conditional probability of any hidden state given the entire sequence, which can be computed by the dot product of forward message and backward message by, i i i i X i;j γt = p(yt = 1jx1:T ) / αtβt = ξt ; (3) j where we define, i;j i j ξt = p(yt = 1; yt−1 = 1; x1:T ); i j / µt−1!t(yt = 1)µt t+1(yt+1 = 1)p(xt+1jyt+1)p(yt+1jyt); i j i = αtβt+1ai;jp(xt+1jyt+1 = 1): The implementation in Matlab can be vectorized by using, i Bt(i) = p(xtjyt = 1); j i A(i; j) = p(yt+1 = 1jyt = 1): 1 2 12 : Conditional Random Fields The relation of those quantities can be simply written in pseudocode as, T αt = (A αt−1): ∗ Bt; βt = A(βt+1: ∗ Bt+1); T ξt = (αt(βt+1: ∗ Bt+1) ): ∗ A; γt = αt: ∗ βt: 1.3 Learning 1.3.1 Supervised Learning The supervised learning is trivial if only we know the true state path.
    [Show full text]
  • The Ising Model
    The Ising Model Today we will switch topics and discuss one of the most studied models in statistical physics the Ising Model • Some applications: – Magnetism (the original application) – Liquid-gas transition – Binary alloys (can be generalized to multiple components) • Onsager solved the 2D square lattice (1D is easy!) • Used to develop renormalization group theory of phase transitions in 1970’s. • Critical slowing down and “cluster methods”. Figures from Landau and Binder (LB), MC Simulations in Statistical Physics, 2000. Atomic Scale Simulation 1 The Model • Consider a lattice with L2 sites and their connectivity (e.g. a square lattice). • Each lattice site has a single spin variable: si = ±1. • With magnetic field h, the energy is: N −β H H = −∑ Jijsis j − ∑ hisi and Z = ∑ e ( i, j ) i=1 • J is the nearest neighbors (i,j) coupling: – J > 0 ferromagnetic. – J < 0 antiferromagnetic. • Picture of spins at the critical temperature Tc. (Note that connected (percolated) clusters.) Atomic Scale Simulation 2 Mapping liquid-gas to Ising • For liquid-gas transition let n(r) be the density at lattice site r and have two values n(r)=(0,1). E = ∑ vijnin j + µ∑ni (i, j) i • Let’s map this into the Ising model spin variables: 1 s = 2n − 1 or n = s + 1 2 ( ) v v + µ H s s ( ) s c = ∑ i j + ∑ i + 4 (i, j) 2 i J = −v / 4 h = −(v + µ) / 2 1 1 1 M = s n = n = M + 1 N ∑ i N ∑ i 2 ( ) i i Atomic Scale Simulation 3 JAVA Ising applet http://physics.weber.edu/schroeder/software/demos/IsingModel.html Dynamically runs using heat bath algorithm.
    [Show full text]
  • Pdf File of Second Edition, January 2018
    Probability on Graphs Random Processes on Graphs and Lattices Second Edition, 2018 GEOFFREY GRIMMETT Statistical Laboratory University of Cambridge copyright Geoffrey Grimmett Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 56 Figures copyright Geoffrey Grimmett Contents Preface ix 1 Random Walks on Graphs 1 1.1 Random Walks and Reversible Markov Chains 1 1.2 Electrical Networks 3 1.3 FlowsandEnergy 8 1.4 RecurrenceandResistance 11 1.5 Polya's Theorem 14 1.6 GraphTheory 16 1.7 Exercises 18 2 Uniform Spanning Tree 21 2.1 De®nition 21 2.2 Wilson's Algorithm 23 2.3 Weak Limits on Lattices 28 2.4 Uniform Forest 31 2.5 Schramm±LownerEvolutionsÈ 32 2.6 Exercises 36 3 Percolation and Self-Avoiding Walks 39 3.1 PercolationandPhaseTransition 39 3.2 Self-Avoiding Walks 42 3.3 ConnectiveConstantoftheHexagonalLattice 45 3.4 CoupledPercolation 53 3.5 Oriented Percolation 53 3.6 Exercises 56 4 Association and In¯uence 59 4.1 Holley Inequality 59 4.2 FKG Inequality 62 4.3 BK Inequalitycopyright Geoffrey Grimmett63 vi Contents 4.4 HoeffdingInequality 65 4.5 In¯uenceforProductMeasures 67 4.6 ProofsofIn¯uenceTheorems 72 4.7 Russo'sFormulaandSharpThresholds 80 4.8 Exercises 83 5 Further Percolation 86 5.1 Subcritical Phase 86 5.2 Supercritical Phase 90 5.3 UniquenessoftheIn®niteCluster 96 5.4 Phase Transition 99 5.5 OpenPathsinAnnuli 103 5.6 The Critical Probability in Two Dimensions 107
    [Show full text]
  • Mean Field Methods for Classification with Gaussian Processes
    Mean field methods for classification with Gaussian processes Manfred Opper Neural Computing Research Group Division of Electronic Engineering and Computer Science Aston University Birmingham B4 7ET, UK. opperm~aston.ac.uk Ole Winther Theoretical Physics II, Lund University, S6lvegatan 14 A S-223 62 Lund, Sweden CONNECT, The Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, 2100 Copenhagen 0, Denmark winther~thep.lu.se Abstract We discuss the application of TAP mean field methods known from the Statistical Mechanics of disordered systems to Bayesian classifi­ cation models with Gaussian processes. In contrast to previous ap­ proaches, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given. 1 Modeling with Gaussian Processes Bayesian models which are based on Gaussian prior distributions on function spaces are promising non-parametric statistical tools. They have been recently introduced into the Neural Computation community (Neal 1996, Williams & Rasmussen 1996, Mackay 1997). To give their basic definition, we assume that the likelihood of the output or target variable T for a given input s E RN can be written in the form p(Tlh(s)) where h : RN --+ R is a priori assumed to be a Gaussian random field. If we assume fields with zero prior mean, the statistics of h is entirely defined by the second order correlations C(s, S') == E[h(s)h(S')], where E denotes expectations 310 M Opper and 0. Winther with respect to the prior. Interesting examples are C(s, s') (1) C(s, s') (2) The choice (1) can be motivated as a limit of a two-layered neural network with infinitely many hidden units with factorizable input-hidden weight priors (Williams 1997).
    [Show full text]
  • MCMC Learning (Slides)
    Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions MCMC Learning Varun Kanade Elchanan Mossel UC Berkeley UC Berkeley August 30, 2013 Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Outline Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Uniform Distribution Learning • Unknown target function f : {−1; 1gn ! {−1; 1g from some class C • Uniform distribution over {−1; 1gn • Random Examples: Monotone Decision Trees [OS06] • Random Walk: DNF expressions [BMOS03] • Membership Query: DNF, TOP [J95] • Main Tool: Discrete Fourier Analysis X Y f (x) = f^(S)χS (x); χS (x) = xi S⊆[n] i2S • Can utilize sophisticated results: hypercontractivity, invariance, etc. • Connections to cryptography, hardness, de-randomization etc. • Unfortunately, too much of an idealization. In practice, variables are correlated. Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Markov Random Fields • Graph G = ([n]; E). Each node takes some value in finite set A. n • Distribution over A : (for φC non-negative, Z normalization constant) 1 Y Pr((σ ) ) = φ ((σ ) ) v v2[n] Z C v v2C clique C Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Markov Random Fields • MRFs widely used in vision, computational biology, biostatistics etc. • Extensive Algorithmic Theory for sampling from MRFs, recovering parameters and structures • Learning Question: Given f : An ! {−1; 1g. (How) Can we learn with respect to MRF distribution? • Can we utilize the structure of the MRF to aid in learning? Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Learning Model • Let M be a MRF with distribution π and f : An ! {−1; 1g the target function • Learning algorithm gets i.i.d.
    [Show full text]
  • Arxiv:1511.03031V2
    submitted to acta physica slovaca 1– 133 A BRIEF ACCOUNT OF THE ISING AND ISING-LIKE MODELS: MEAN-FIELD, EFFECTIVE-FIELD AND EXACT RESULTS Jozef Streckaˇ 1, Michal Jasˇcurˇ 2 Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. Saf´arikˇ University, Park Angelinum 9, 040 01 Koˇsice, Slovakia The present article provides a tutorial review on how to treat the Ising and Ising-like models within the mean-field, effective-field and exact methods. The mean-field approach is illus- trated on four particular examples of the lattice-statistical models: the spin-1/2 Ising model in a longitudinal field, the spin-1 Blume-Capel model in a longitudinal field, the mixed-spin Ising model in a longitudinal field and the spin-S Ising model in a transverse field. The mean- field solutions of the spin-1 Blume-Capel model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A con- tinuous quantum phase transition of the spin-S Ising model driven by a transverse magnetic field is also explored within the mean-field method. The effective-field theory is elaborated within a single- and two-spin cluster approach in order to demonstrate an efficiency of this ap- proximate method, which affords superior approximate results with respect to the mean-field results. The long-standing problem of this method concerned with a self-consistent deter- mination of the free energy is also addressed in detail. More specifically, the effective-field theory is adapted for the spin-1/2 Ising model in a longitudinal field, the spin-S Blume-Capel model in a longitudinal field and the spin-1/2 Ising model in a transverse field.
    [Show full text]
  • Statistical Field Theory University of Cambridge Part III Mathematical Tripos
    Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2017 Statistical Field Theory University of Cambridge Part III Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/sft.html [email protected] –1– Recommended Books and Resources There are a large number of books which cover the material in these lectures, although often from very di↵erent perspectives. They have titles like “Critical Phenomena”, “Phase Transitions”, “Renormalisation Group” or, less helpfully, “Advanced Statistical Mechanics”. Here are some that I particularly like Nigel Goldenfeld, Phase Transitions and the Renormalization Group • Agreatbook,coveringthebasicmaterialthatwe’llneedanddelvingdeeperinplaces. Mehran Kardar, Statistical Physics of Fields • The second of two volumes on statistical mechanics. It cuts a concise path through the subject, at the expense of being a little telegraphic in places. It is based on lecture notes which you can find on the web; a link is given on the course website. John Cardy, Scaling and Renormalisation in Statistical Physics • Abeautifullittlebookfromoneofthemastersofconformalfieldtheory.Itcoversthe material from a slightly di↵erent perspective than these lectures, with more focus on renormalisation in real space. Chaikin and Lubensky, Principles of Condensed Matter Physics • Shankar, Quantum Field Theory and Condensed Matter • Both of these are more all-round condensed matter books, but with substantial sections on critical phenomena and the renormalisation group. Chaikin and Lubensky is more traditional, and packed full of content. Shankar covers modern methods of QFT, with an easygoing style suitable for bedtime reading. Anumberofexcellentlecturenotesareavailableontheweb.Linkscanbefoundon the course webpage: http://www.damtp.cam.ac.uk/user/tong/sft.html.
    [Show full text]
  • Markov Random Fields: – Geman, Geman, “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images”, IEEE PAMI 6, No
    MarkovMarkov RandomRandom FieldsFields withwith ApplicationsApplications toto MM--repsreps ModelsModels Conglin Lu Medical Image Display and Analysis Group University of North Carolina, Chapel Hill MarkovMarkov RandomRandom FieldsFields withwith ApplicationsApplications toto MM--repsreps ModelsModels Outline: Background; Definition and properties of MRF; Computation; MRF m-reps models. MarkovMarkov RandomRandom FieldsFields Model a large collection of random variables with complex dependency relationships among them. MarkovMarkov RandomRandom FieldsFields • A model based approach; • Has been applied to a variety of problems: - Speech recognition - Natural language processing - Coding - Image analysis - Neural networks - Artificial intelligence • Usually used within the Bayesian framework. TheThe BayesianBayesian ParadigmParadigm X = space of the unknown variables, e.g. labels; Y = space of data (observations), e.g. intensity values; Given an observation y∈Y, want to make inference about x∈X. TheThe BayesianBayesian ParadigmParadigm Prior PX : probability distribution on X; Likelihood PY|X : conditional distribution of Y given X; Statistical inference is based on the posterior distribution PX|Y ∝ PX •PY|X . TheThe PriorPrior DistributionDistribution • Describes our assumption or knowledge about the model; • X is usually a high dimensional space. PX describes the joint distribution of a large number of random variables; • How do we define PX? MarkovMarkov RandomRandom FieldsFields withwith ApplicationsApplications toto MM--repsreps ModelsModels Outline: 9 Background; Definition and properties of MRF; Computation; MRF m-reps models. AssumptionsAssumptions •X = {Xs}s∈S, where each Xs is a random variable; S is an index set and is finite; • There is a common state space R:Xs∈R for all s ∈ S; | R | is finite; • Let Ω = {ω=(x , ..., x ): x ∈ , 1≤i≤N} be s1 sN si R the set of all possible configurations.
    [Show full text]
  • Anomaly Detection Based on Wavelet Domain GARCH Random Field Modeling Amir Noiboar and Israel Cohen, Senior Member, IEEE
    IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 5, MAY 2007 1361 Anomaly Detection Based on Wavelet Domain GARCH Random Field Modeling Amir Noiboar and Israel Cohen, Senior Member, IEEE Abstract—One-dimensional Generalized Autoregressive Con- Markov noise. It is also claimed that objects in imagery create a ditional Heteroscedasticity (GARCH) model is widely used for response over several scales in a multiresolution representation modeling financial time series. Extending the GARCH model to of an image, and therefore, the wavelet transform can serve as multiple dimensions yields a novel clutter model which is capable of taking into account important characteristics of a wavelet-based a means for computing a feature set for input to a detector. In multiscale feature space, namely heavy-tailed distributions and [17], a multiscale wavelet representation is utilized to capture innovations clustering as well as spatial and scale correlations. We periodical patterns of various period lengths, which often ap- show that the multidimensional GARCH model generalizes the pear in natural clutter images. In [12], the orientation and scale casual Gauss Markov random field (GMRF) model, and we de- selectivity of the wavelet transform are related to the biological velop a multiscale matched subspace detector (MSD) for detecting anomalies in GARCH clutter. Experimental results demonstrate mechanisms of the human visual system and are utilized to that by using a multiscale MSD under GARCH clutter modeling, enhance mammographic features. rather than GMRF clutter modeling, a reduced false-alarm rate Statistical models for clutter and anomalies are usually can be achieved without compromising the detection rate. related to the Gaussian distribution due to its mathematical Index Terms—Anomaly detection, Gaussian Markov tractability.
    [Show full text]
  • Lecture 8: the Ising Model Introduction
    Lecture 8: The Ising model Introduction I Up to now: Toy systems with interesting properties (random walkers, cluster growth, percolation) I Common to them: No interactions I Add interactions now, with significant role I Immediate consequence: much richer structure in model, in particular: phase transitions I Simulate interactions with RNG (Monte Carlo method) I Include the impact of temperature: ideas from thermodynamics and statistical mechanics important I Simple system as example: coupled spins (see below), will use the canonical ensemble for its description The Ising model I A very interesting model for understanding some properties of magnetic materials, especially the phase transition ferromagnetic ! paramagnetic I Intrinsically, magnetism is a quantum effect, triggered by the spins of particles aligning with each other I Ising model a superb toy model to understand this dynamics I Has been invented in the 1920's by E.Ising I Ever since treated as a first, paradigmatic model The model (in 2 dimensions) I Consider a square lattice with spins at each lattice site I Spins can have two values: si = ±1 I Take into account only nearest neighbour interactions (good approximation, dipole strength falls off as 1=r 3) I Energy of the system: P E = −J si sj hiji I Here: exchange constant J > 0 (for ferromagnets), and hiji denotes pairs of nearest neighbours. I (Micro-)states α characterised by the configuration of each spin, the more aligned the spins in a state α, the smaller the respective energy Eα. I Emergence of spontaneous magnetisation (without external field): sufficiently many spins parallel Adding temperature I Without temperature T : Story over.
    [Show full text]
  • Probability on Graphs Random Processes on Graphs and Lattices
    Probability on Graphs Random Processes on Graphs and Lattices GEOFFREY GRIMMETT Statistical Laboratory University of Cambridge c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 44 Figures c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Contents Preface ix 1 Random walks on graphs 1 1.1 RandomwalksandreversibleMarkovchains 1 1.2 Electrical networks 3 1.3 Flowsandenergy 8 1.4 Recurrenceandresistance 11 1.5 Polya's theorem 14 1.6 Graphtheory 16 1.7 Exercises 18 2 Uniform spanning tree 21 2.1 De®nition 21 2.2 Wilson's algorithm 23 2.3 Weak limits on lattices 28 2.4 Uniform forest 31 2.5 Schramm±LownerevolutionsÈ 32 2.6 Exercises 37 3 Percolation and self-avoiding walk 39 3.1 Percolationandphasetransition 39 3.2 Self-avoiding walks 42 3.3 Coupledpercolation 45 3.4 Orientedpercolation 45 3.5 Exercises 48 4 Association and in¯uence 50 4.1 Holley inequality 50 4.2 FKGinequality 53 4.3 BK inequality 54 4.4 Hoeffdinginequality 56 c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 vi Contents 4.5 In¯uenceforproductmeasures 58 4.6 Proofsofin¯uencetheorems 63 4.7 Russo'sformulaandsharpthresholds 75 4.8 Exercises 78 5 Further percolation 81 5.1 Subcritical phase 81 5.2 Supercritical phase 86 5.3 Uniquenessofthein®nitecluster 92 5.4 Phase transition 95 5.5 Openpathsinannuli 99 5.6 The critical probability in two dimensions 103 5.7 Cardy's formula 110 5.8 The
    [Show full text]